Extreme ratio between spectral and Frobenius norms of nonnegative tensors

TL;DR

This paper investigates the ratio between spectral and Frobenius norms of nonnegative tensors, establishing bounds, characterizing extreme tensors, and comparing with nuclear norms, thus advancing understanding in multilinear algebra.

Contribution

It provides a tight lower bound for the spectral to Frobenius norm ratio in nonnegative tensors and characterizes the conditions for extremal tensors, extending known results to this class.

Findings

Tight lower bound achieved by a broad class of nonnegative tensors

Asymptotic order of magnitude determined for symmetric and general tensors

The ratio differs from the Frobenius to nuclear norm ratio in nonnegative tensors

Abstract

One of the fundamental problems in multilinear algebra, the minimum ratio between the spectral and Frobenius norms of tensors, has received considerable attention in recent years. While most values are unknown for real and complex tensors, the asymptotic order of magnitude and tight lower bounds have been established. However, little is known about nonnegative tensors. In this paper, we present an almost complete picture of the ratio for nonnegative tensors. In particular, we provide a tight lower bound that can be achieved by a wide class of nonnegative tensors under a simple necessary and sufficient condition, which helps to characterize the extreme tensors and obtain results such as the asymptotic order of magnitude. We show that the ratio for symmetric tensors is no more than that for general tensors multiplied by a constant depending only on the order of tensors, hence determining the asymptotic order of magnitude for real, complex, and nonnegative symmetric tensors. We also find that the ratio is in general different to the minimum ratio between the Frobenius and nuclear norms for nonnegative tensors, a sharp contrast to the case for real tensors and complex tensors.